Question

Q1 Electric cars of the same type  have ranges, X, that are normally distributed with a mean of 340 km and a standard deviation of 20 km, when driven on a test track. Define

(a)  Find L and U such that

(b) Find the probability that the range on the test track of a randomly chosen car of this type is between 300 km and 350 km.

(c) Given that , what range on the test track can the manufacturer claim that 99% of cars of this type will exceed? 

Q2 A battery for an astronomical instrument in the capsule in a space flight has a lifetime, X, that is normally distributed with a mean of 100 hours and a standard deviation of 30 hours. 

(a) What is the mean and standard deviation of the distribution of the total lifetime, T that is , of three batteries used consecutively, if the lifetimes are independently distributed? 

(b) What is the probability that three batteries will suffice for a mission of 200 hours? 

(c) Now suppose that the length of the mission is independently normally distributed with a mean of 200 hours and a standard deviation of 50 hours. 

What is the probability that three batteries will suffice for the mission?

Q3 A measurement M of the deviation D of the gold content in a particular bottle from the mean of all such bottles has a measurement error E. That is:

Now assume that D and E have means of 0 and variances respectively. Also assume that D and E are independent.

(a) Express the variance of M, , in terms of the variances of D and E.

(b) Write

and hence find an expression for the covariance of M and E, and so the correlation between M and E. 

Q4 The lifetime of a component, T, has a Weibull distribution of the form

(a) Give an expression for the probability that T exceeds 1 and is less than 2, in terms of a.

(b) Give an expression for the probability that T is less than 2 given that it exceeds 1, in terms of a?

(c) Now suppose a equals 1. What is the probability that T is less than 1. What is the probability that T is less than 2 given that it exceeds 1?

Q5 The arsenic levels in water from a random sample of 16 wells in a certain region were found to have a mean of 63.1 micrograms per litre (ppb). [The WHO provisional guideline for arsenic in drinking water is that it should be below 10 ppb, though it is recognised that at least 140 million in 50 different countries drink water above this limit.]

(a) Construct an approximate 95% confidence interval for the mean in the corresponding population if the population standard deviation is assumed to be 40 ppb. 

(b) Construct an approximate 95% confidence interval for the mean in the corresponding population using the sample standard deviation of 38.5 ppb.

Answer 1

The answer for above problem is explained below.